A quadrilateral, sometimes also known as a tetragon or quadrangle (Johnson 1929, p. 61), is a four-sided polygon. If not explicitly stated, all four polygon vertices are generally taken to lie in a plane. (If the points do not lie in a plane, the quadrilateral is called a skew quadrilateral.) There are three topological types of quadrilaterals (Wenninger 1983, p. 50): convex quadrilaterals (left figure), concave quadrilaterals (middle figure), and crossed quadrilaterals (or butterflies, or bow-ties; right figure).
A quadrilateral with two sides parallel is called a trapezoid, whereas a quadrilateral with opposite pairs of sides parallel is called a parallelogram.
For a planar convex quadrilateral (left figure above), let the lengths of the sides be 
, 
, 
, and 
, the semiperimeter 
, and the polygon diagonals 
and 
. The polygon diagonals
are perpendicular iff 
.
The following table compares selected classes of convex quadrilaterals (Josefsson 2016, p. 78; Josefsson 2017, p. 214). Each drawing is one representative, and the listed properties hold for every member of the class. A blank makes no assertion. The parallel-pair count refers to pairs of opposite parallel sides, and C and I denote a guaranteed circumcircle and incircle, respectively.
For this table, label the vertices 
, 
, 
, and 
in boundary order, with 
, 
, 
, 
, 
, and 
. The interior
angles at these vertices are 
, 
, 
, and 
, respectively. If the diagonals
meet at 
, write 
, 
, 
, and 
. For the trapezoid
rows, take 
, and for the kite
row, take 
and 
.
Black segments show each quadrilateral and its diagonals. Blue circles, where present, are the circumcircle or incircle indicated by C and I in the table.
| representative | class | parallel pairs | sides | angles | diagonals | circles |
| square | 2 |  | 
each |  ,  ,  ,  | C, I |
| rectangle | 2 |  ,  |  each |  ,  ,  | C |
| rhombus | 2 |  |  ,  |  ,  ,  | I |
| parallelogram | 2 |  ,  |  ,  |  ,  | |
| trapezoid |  |  | |||
| isosceles trapezoid |  |  |  ,  |  | C |
| kite |  ,  |  |  ,  | I | |
| cyclic |  |  | C | ||
| tangential |  | I | |||
| bicentric |  |  |  | C, I | |
| orthodiagonal |  |  | |||
| equidiagonal |  | ||||
| midsquare |  ,  | ||||
| bisect-diagonal |  or  |
These classes overlap and use inclusive definitions. For example, a square is both a rectangle and a rhombus, and a rectangle is both a parallelogram and an isosceles trapezoid. More systematic classifications prescribe equalities among side lengths, interior angles, and the four line segments determined by the intersection of the diagonals. Dalcín (2022, pp. 272-274) organizes 35 convex families using 65 definitions without redundant conditions based on such equalities.
An equation for the sum of the squares of side lengths is
 |
(1)
|
where 
is the length of the line
segment joining the midpoints of the polygon
diagonals (Casey 1888, p. 22).
For bicentric quadrilaterals, the circumcircle and incircle satisfy
 |
(2)
|
where 
is the circumradius, 
is the inradius, and 
is the separation of centers.
Given any five points in the plane in general position, four will form a convex quadrilateral. This result is a special case of the so-called happy end problem (Hoffman 1998, pp. 74-78).
The quadrilateral half-angle formulas relate the side lengths of a planar convex quadrilateral to combinations of half angles at opposite vertices.
The area 
of a planar convex quadrilateral
can be expressed in terms of the vectors corresponding
to its two polygon diagonals. Represent the sides
of the quadrilateral by the vectors 
, 
, 
, and 
arranged such that 
and the polygon diagonals
by the vectors 
and 
arranged so that 
and 
. Then
 |  |  |
(3)
|
 |  |  |
(4)
|
where 
is the determinant and 
is a two-dimensional cross
product.
Other formulas express the area of a planar convex quadrilateral in terms of its side and polygon diagonal lengths, including
 |  |  |
(5)
|
 |  |  |
(6)
|
(Beyer 1987, p. 123), Bretschneider's formula
 |  |  |
(7)
|
 |  |  |
(8)
|
(Coolidge 1939; Ivanoff 1960; Beyer 1987, p. 123) where 
is the semiperimeter, and
![K=sqrt((s-a)(s-b)(s-c)(s-d)-abcdcos^2[1/2(A+B)])](/images/equations/Quadrilateral/NumberedEquation3.svg) |
(9)
|
(Bretschneider 1842; Strehlke 1842; Coolidge 1939; Beyer 1987, p. 123).
The geometric centroid of the polygon vertices of a quadrilateral occurs at the point of
intersection of the bimedians
(i.e., the line segments 
and 
joining pairs of opposite midpoints)
(Honsberger 1995, pp. 36-37). In addition, it is the midpoint
of the line segment 
connecting the midpoints
of the polygon diagonals 
and 
(Honsberger 1995, pp. 39-40).
The four angle bisectors of a quadrilateral intersect adjacent bisectors in four concyclic points (Honsberger 1995, p. 35).
Any non-self-intersecting quadrilateral tiles the plane.
There is a relationship between the six distances 
, 
, 
, 
, 
, and 
between the four points of
a quadrilateral (Weinberg 1972):
 |
(10)
|
This can be most simply derived by setting the left side of the Cayley-Menger determinant
 |
(11)
|
equal to 0 (corresponding to a tetrahedron of volume 0), thus giving a relationship between the distances between polygon vertices of a planar quadrilateral (Uspensky 1948, p. 256).
A special type of quadrilateral is the cyclic quadrilateral, for which a circle can be circumscribed so that it touches each polygon vertex. Another special type is a tangential quadrilateral, for which a circle can be inscribed so it is tangent to each polygon edge. A quadrilateral that is both cyclic and tangential is called a bicentric quadrilateral. A convex quadrilateral is a bielliptic quadrilateral when its unique inellipse and circumellipse of minimum eccentricity have the same eccentricity (Horwitz 2010).
